Optimal. Leaf size=273 \[ \frac{b \tan (c+d x) \left (a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac{\left (24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec (c+d x) \left (a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right )}{24 d}+a^3 x (a B+4 A b)-\frac{b \tan (c+d x) (12 a A-7 a C-4 b B) (a+b \sec (c+d x))^2}{12 d}-\frac{b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]
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Rubi [A] time = 0.582644, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4094, 4056, 4048, 3770, 3767, 8} \[ \frac{b \tan (c+d x) \left (a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 d}+\frac{\left (24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec (c+d x) \left (a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right )}{24 d}+a^3 x (a B+4 A b)-\frac{b \tan (c+d x) (12 a A-7 a C-4 b B) (a+b \sec (c+d x))^2}{12 d}-\frac{b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^4}{d} \]
Antiderivative was successfully verified.
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Rule 4094
Rule 4056
Rule 4048
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\int (a+b \sec (c+d x))^3 \left (4 A b+a B+(b B+a C) \sec (c+d x)-b (4 A-C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{4} \int (a+b \sec (c+d x))^2 \left (4 a (4 A b+a B)+\left (4 A b^2+8 a b B+4 a^2 C+3 b^2 C\right ) \sec (c+d x)-b (12 a A-4 b B-7 a C) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}-\frac{b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{12} \int (a+b \sec (c+d x)) \left (12 a^2 (4 A b+a B)+\left (36 a^2 b B+8 b^3 B+12 a^3 C+a b^2 (36 A+23 C)\right ) \sec (c+d x)+b \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac{b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{24} \int \left (24 a^3 (4 A b+a B)+3 \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \sec (c+d x)+4 b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \sec ^2(c+d x)\right ) \, dx\\ &=a^3 (4 A b+a B) x+\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac{b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}+\frac{1}{6} \left (b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right )\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{8} \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 (4 A b+a B) x+\frac{\left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac{b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}-\frac{\left (b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d}\\ &=a^3 (4 A b+a B) x+\frac{\left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{A (a+b \sec (c+d x))^4 \sin (c+d x)}{d}+\frac{b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \tan (c+d x)}{6 d}+\frac{b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}-\frac{b (12 a A-4 b B-7 a C) (a+b \sec (c+d x))^2 \tan (c+d x)}{12 d}-\frac{b (4 A-C) (a+b \sec (c+d x))^3 \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [B] time = 6.89277, size = 813, normalized size = 2.98 \[ \frac{\left (-8 C a^4-32 b B a^3-48 A b^2 a^2-24 b^2 C a^2-16 b^3 B a-4 A b^4-3 b^4 C\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^6(c+d x)}{4 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\left (8 C a^4+32 b B a^3+48 A b^2 a^2+24 b^2 C a^2+16 b^3 B a+4 A b^4+3 b^4 C\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) (a+b \sec (c+d x))^4 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \cos ^6(c+d x)}{4 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x))^4 \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (36 B (c+d x) a^4+48 B (c+d x) \cos (2 (c+d x)) a^4+12 B (c+d x) \cos (4 (c+d x)) a^4+12 A \sin (c+d x) a^4+18 A \sin (3 (c+d x)) a^4+6 A \sin (5 (c+d x)) a^4+144 A b (c+d x) a^3+192 A b (c+d x) \cos (2 (c+d x)) a^3+48 A b (c+d x) \cos (4 (c+d x)) a^3+96 b C \sin (2 (c+d x)) a^3+48 b C \sin (4 (c+d x)) a^3+72 b^2 C \sin (c+d x) a^2+144 b^2 B \sin (2 (c+d x)) a^2+72 b^2 C \sin (3 (c+d x)) a^2+72 b^2 B \sin (4 (c+d x)) a^2+48 b^3 B \sin (c+d x) a+96 A b^3 \sin (2 (c+d x)) a+128 b^3 C \sin (2 (c+d x)) a+48 b^3 B \sin (3 (c+d x)) a+48 A b^3 \sin (4 (c+d x)) a+32 b^3 C \sin (4 (c+d x)) a+12 A b^4 \sin (c+d x)+33 b^4 C \sin (c+d x)+32 b^4 B \sin (2 (c+d x))+12 A b^4 \sin (3 (c+d x))+9 b^4 C \sin (3 (c+d x))+8 b^4 B \sin (4 (c+d x))\right ) \cos ^2(c+d x)}{48 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 457, normalized size = 1.7 \begin{align*}{\frac{A{a}^{4}\sin \left ( dx+c \right ) }{d}}+B{a}^{4}x+{\frac{B{a}^{4}c}{d}}+{\frac{{a}^{4}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{a}^{3}Abx+4\,{\frac{A{a}^{3}bc}{d}}+4\,{\frac{B{a}^{3}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{{a}^{3}bC\tan \left ( dx+c \right ) }{d}}+6\,{\frac{A{a}^{2}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+6\,{\frac{{a}^{2}{b}^{2}B\tan \left ( dx+c \right ) }{d}}+3\,{\frac{C{a}^{2}{b}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+3\,{\frac{C{a}^{2}{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+4\,{\frac{Aa{b}^{3}\tan \left ( dx+c \right ) }{d}}+2\,{\frac{a{b}^{3}B\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+2\,{\frac{a{b}^{3}B\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{8\,Ca{b}^{3}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{4\,Ca{b}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{A{b}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{2\,B{b}^{4}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{B{b}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{C{b}^{4}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,C{b}^{4}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,C{b}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08799, size = 582, normalized size = 2.13 \begin{align*} \frac{48 \,{\left (d x + c\right )} B a^{4} + 192 \,{\left (d x + c\right )} A a^{3} b + 64 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a b^{3} + 16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b^{4} - 3 \, C b^{4}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, C a^{2} b^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, B a b^{3}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, A b^{4}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{4}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, B a^{3} b{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, A a^{2} b^{2}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, A a^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 288 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.629816, size = 725, normalized size = 2.66 \begin{align*} \frac{48 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d x \cos \left (d x + c\right )^{4} + 3 \,{\left (8 \, C a^{4} + 32 \, B a^{3} b + 24 \,{\left (2 \, A + C\right )} a^{2} b^{2} + 16 \, B a b^{3} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (8 \, C a^{4} + 32 \, B a^{3} b + 24 \,{\left (2 \, A + C\right )} a^{2} b^{2} + 16 \, B a b^{3} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 6 \, C b^{4} + 16 \,{\left (6 \, C a^{3} b + 9 \, B a^{2} b^{2} + 2 \,{\left (3 \, A + 2 \, C\right )} a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (24 \, C a^{2} b^{2} + 16 \, B a b^{3} +{\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.45465, size = 1134, normalized size = 4.15 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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